1. Quantum Physics
Comes of Age I
incident II
transmitted
reflected

5. Dispute about the interpretation of QM
Einstein
Bohr
”God does NOT play dice”
”God plays dice”
Dispute about the interpretation of QM

6. Copenhagen school
Bohr, Heisenberg, Pauli

7. Erwin
Schrödinger
“The task is not so much to see what no-one has yet seen, but to think what nobody has yet thought, about that which everybody sees.”

8. Quantum Mechanics: Schrodinger's Cat: History
First arose in 1935
Described in letters exchanged between Schrodinger and Einstein 
Both were rallying against the Copenhagen Interpretation where "wave function collapse" occurs only at observation
Einstein considered a power keg that is paradoxically both exploded and un-exploded
Schrodinger agreed, and replied by describing the famous cat paradox
Schrodinger did not expect that the experiment would ever be done -- it was considered a "reduction to absurdity" argument against the Copenhagen Interpretation of quantum mechanics

9. The Thought Experiment
A steel chamber

10. The Thought Experiment (contd.)
A cat is placed in it

11. The Thought Experiment (contd.)
And with it, a small amount of a radioactive substance

12. The Thought Experiment (contd.)
And a veil of poison
The box is shielded against quantum decoherence

13. The Thought Experiment (contd.)
The cat can be alive 

14. The Thought Experiment (contd.)
The cat can be dead 

15. The Thought Experiment (contd.)
Or both?

16. The Thought Experiment (contd.)
Or both? Yes 

17. How ??
Any atom in the radioactive substance decays The veil breaks open and releases poison The cat dies

18. How ??
No atom in the radioactive substance decays The poison is not released The cat is alive

19. How ??
Atom in the radioactive substance is in a state of both ‘decayed’ and ‘undecayed’ The cat is both dead and alive

20. The Principle of Superposition
The principle of superposition says that if an object can be in any configuration, and if the object could also be in another configuration, then the object can also be in a state which is a superposition of the two.
The thought experiment uses this and transforms an indeterminacy in the atomic domain to macroscopic indeterminacy.

21. The Problem
Thus an atom of the radioactive substance can be in both decayed and non-decayed states at the same time.
This leads to the conclusion that the cat can be both dead and alive at the same time.
But we can never observe the cat to be both dead and alive at the same time!!!

22. an interesting explanation - The Copenhagen Interpretation
A system stops being in a superposition of states and becomes either one or the other when an observation takes place.

23. Quantum Mechanics: Schrodinger's Cat: Description
Cat in a closed box 
A quantum decision is made in the box that may kill the cat
Time passes

24. Quantum Mechanics: Schrodinger's Cat: Description
Just before the box is opened: is the cat dead? 1. Either yes or no, but you won't know until the box is opened. 2. Both yes and no until the box is opened, then either yes or no.

25. Schrodinger's Cat: Copenhagen Interpretation
2.  Both yes and no until the box is opened, then either yes or no.
The cat is BOTH alive and dead until the box is opened!
Cat's "wave function state" collapses only when the box is opened
"Opening the box" can really mean
actually opening the box
looking into the box
doing any determinative experiment to the closed box

26. Schrodinger's Cat Contrasting Interpretations
1.  Either yes or no, but you won't know until the box is opened.
Many Worlds Interpretation
separate universes house dead and alive cats
these universes are decoherent -- do not interact
Ensemble Interpretation
individual cats are either alive or dead, not both, but you can't know which until the box is opened
statistics are only built up when many single-cat systems are observed

28. Heisenberg’s Matrix Mechanics
The Schrödinger Equation
Heisenberg’s Matrix Mechanics
1924: de Broglie suggests particles are waves
Mid-1925: Werner Heisenberg introduces Matrix Mechanics
Semi-philosophical, it only considers observable quantities
It used matrices, which were not that familiar at the time
It refused to discuss what happens between measurements
In 1927 he derives uncertainty principles
Late 1925: Erwin Schrödinger proposes wave mechanics
Used waves, more familiar to scientists at the time
Initially, Heisenberg’s and Schrödinger’s formulations were competing
Eventually, Schrödinger showed they were equivalent; different descriptions which produced the same predictions
Both formulations are used today, but Schrödinger is easier to understand

29. The Free Schrödinger Equation
1925: Erwin Schrödinger proposes wave mechanics
Peter Debye suggested to him he needed to find a wave equation for quantum mechanics
He hit on the idea of using complex waves
The rest is history
Starting point: Energy/Momentum relationship
Multiply by the wave function on the right
Use de Broglie relations to rewrite
Use relationships for complex waves to rewrite with derivatives

30. Sample Problem with Free Schrödinger
Show that the following expression satisfies the free Schrödinger equation, and find the constant A:

31. Sample Problem with Free Schrödinger (2)
Show that the following expression satisfies the free Schrödinger equation, and find the constant A:
Multiply by 2mt5/2/

32. The Schrödinger Equation
The General Prescription for Classical  Quantum:
Write a formula for the energy in terms of momentum and position
Transform Energy and momentum using the following prescription:
Rewrite it as a wave equation
What if we have forces?
Need to add potential energy V(x,t) on top of kinetic energy term

33. Comments on Schrödinger Equation
1. This equation is inherently complex
You MUST use complex wave functions
2. This equation is first order in time
It has only first derivatives with respect to time
If you know the value at t = 0, you can work it out at subsequent times
Proved using Taylor expansion:
Initial conditions:
Classical physics
x(t = 0) and v(t = 0)
Initial conditions:
Quantum physics
(x,t = 0)

34. The Superposition Principle
3. This equation is linear
The wave function appears to the first power everywhere
You can take linear combinations of solutions:
Let 1 and 2 be two solutions of Schrödinger. Then so is
where c1 and c2 are arbitrary complex numbers
Q.E.D

35. Time Independent problems
Often [usually] the potential does not depend on time: V = V(x).
To solve this equation, we try separation of variable:
Plug this guess in:
Divide by the original wave function
Note that left side is independent of x, and right side is independent of t.
Both sides must be independent of both x and t
Both sides must be equal to a constant, called E (the energy)

36. Solving the time equation
We have turned one equation into two
But the two equations are now ordinary differential equations
Furthermore, the first equation is easy to solve:
These types of solutions are called stationary states
Why? Don’t they have time in them?
The probability density is independent of time

37. The Time Independent Schrödinger Eqn
Multiply by (x) again
This equation is much easier to solve than the original equation
ODE’s are easier than PDE’s
It can pretty easily be solved numerically, if necessary
Note that it is a real equation – you don’t need complex numbers
Imagine finding all possible solutions n(x) with energy En
Then we can find solutions of the original Schrödinger Equation
The most general solution is superposition of this solution